---
tag: ZP55RF3A0C
kind: concept
name: Uniform algebra
msc: [46J10, 46J05]
sources:
  - url: https://en.wikipedia.org/wiki/Uniform_algebra
same_as:
  wikidata: Q7885096
  wikipedia: Uniform algebra
---

## Summary

A uniform algebra is a closed subalgebra of the continuous complex functions on a compact Hausdorff space that contains the constants and separates points.
It is a commutative [[Banach algebra|9KYWN94QDN]] with the sup norm.
The main examples come from complex analysis, such as the disc algebra of functions holomorphic inside the disc and continuous up to the boundary.
The [[Stone–Weierstrass theorem|1P8D3MK79H]] says the only uniform algebra on $X$ closed under complex conjugation is $C(X)$ itself.
So the theory of uniform algebras studies the closed subalgebras that are not closed under conjugation.

## Explanations

### L1 human

Start with all continuous functions on a compact space, measured by their largest absolute value.
A uniform algebra is a smaller family that is still closed under sums, products and uniform limits, contains the constants, and can tell any two points apart.
The polynomials in $z$ on the closed unit disc, together with all their uniform limits, form such a family.
It is smaller than all continuous functions: $\overline{z}$ is not a uniform limit of polynomials in $z$.